From von Mises' Impossibility of a Gambling System to Probabilistic Martingales
Francesca Zaffora Blando
Abstract:
Algorithmic randomness draws on computability theory to offer rigorous formulations of the notion of randomness for mathematical objects. In addition to having evolved into a highly technical branch of mathematical logic, algorithmic randomness prompts numerous methodological questions. This thesis aims at addressing some of these questions, together with some of the technical challenges that they spawn. In the first part, we discuss the work on randomness and the foundations of probability of the Austrian mathematician Richard von Mises [1919], whose theory of collectives constitutes the first attempt at providing a formal definition of randomness. Our main objective there is to ascertain the reasons that led to the demise of von Mises’ approach in favour of algorithmic randomness. Then, we turn to the myriad definitions of randomness that have been proposed within the algorithmic paradigm, and we focus on the issue of whether any of these definitions can be said to be more legitimate than the others. In particular, we consider some of the objections that have been levelled against Martin-Löf randomness [1966] (arguably, the most popular notion of algorithmic randomness in the literature), concentrating on the famous critique of Martin-Löf randomness due to Schnorr [1971a] and on a more recent critique due Osherson and Weinstein [2008], which relies on a learning-theoretic argument. We point out the inconclusiveness of these criticisms, and we recommend a pluralistic approach to algorithmic randomness. While appraising Osherson and Weinstein’s critique, we also allow ourselves a brief learning- theoretic digression and further study the notion of Kurtz randomness in learning-theoretic terms. In light of the increasing amount of attention being paid to Schnorr’s critique of Martin-Löf randomness in the literature, in the second part of this thesis we consider some of the technical implications of taking said critique seriously. In their paper on probabilistic algorithmic randomness [2013], Buss and Minnes countenance Schnorr’s critique by offering a characterisation of Martin-Löf randomness in terms of computable probabilistic martingales (betting strategies). Buss and Minnes also ask whether there are any natural conditions on the class of probabilistic martingales that can be used to characterise other common algorithmic randomness notions. We answer their question in the affirmative both in the monotonic and the non-monotonic setting, by providing probabilistic characterisations of Martin-Löf randomness, Schnorr randomness, Kurtz randomness and Kolmogorov-Loveland randomness.